Combinatorial description around any vertex of a cubical n-manifold

Abstract

We say that a topological space N is a cubical n-manifold if it is a topological manifold of dimension n contained in the n-skeleton of the canonical cubulation of Rn+2. For instance, any smooth n-knot Sn → Rn+2 can be deformed by an ambient isotopy into a cubical n-knot. An open question is the following: Is any closed, oriented, cubical n-manifold N in Rn+2, n > 2, smoothable? If the response is positive, we could give a discrete description of any smooth n-manifold; in particular, if we can stablish that for smooth n-knots, that fact can be useful to define invariants. One of the main dificulties to answer the above question lies in the understanding of how N looks at each vertex of the canonical cubulation. In this paper, we analyze all possible combinatorial behaviors around any vertex of any cubical manifold of dimension n, via the study of the cycles on the complete graph K2n.